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Robust Feedback Stability Analysis for Systems with Simultaneous Gain and Phase Uncertainty


Core Concepts
The core message of this article is to derive less conservative sufficient conditions for the robust feedback stability of linear time-invariant systems involving sectored-disk uncertainty, which encompasses simultaneous gain and phase constraints on the uncertain dynamics.
Abstract
The article investigates the robust feedback stability problem for multiple-input-multiple-output linear time-invariant systems with sectored-disk uncertainty, which refers to dynamic uncertainty subject to simultaneous gain and phase constraints. Key highlights: The authors leverage the notion of the Davis-Wielandt (DW) shell, a higher-dimensional generalization of the numerical range, to characterize the shape of the DW shell union of sectored-disk matrices. Based on the DW shell analysis, the authors derive a fundamental static matrix problem that serves as a key component in addressing the feedback stability. A sufficient condition and a necessary condition for the matrix sectored-disk problem are established, providing a less conservative approach compared to using the small gain theorem and the small phase theorem alone. Several linear matrix inequality-based conditions are developed for efficient computation and verification of feedback robust stability against sectored-disk uncertainty. The article provides a comprehensive analysis of the robust stability problem involving simultaneous gain and phase constraints, offering insights into the interplay between norm and phase information in feedback systems.
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Key Insights Distilled From

by Jiajin Liang... at arxiv.org 04-09-2024

https://arxiv.org/pdf/2404.05609.pdf
Feedback Stability Under Mixed Gain and Phase Uncertainty

Deeper Inquiries

How can the proposed robust stability conditions be extended to address more general forms of structured uncertainty beyond the sectored-disk model

The proposed robust stability conditions for the sectored-disk model can be extended to address more general forms of structured uncertainty by considering a broader range of constraints on the norm and phase of the uncertain matrices. One approach could involve incorporating additional constraints on the singular values or eigenvalues of the matrices, allowing for a more comprehensive characterization of the uncertainty set. By formulating suitable conditions based on these extended constraints, the robust stability analysis can be enhanced to accommodate a wider variety of uncertainties in practical systems. Additionally, exploring the interplay between different types of uncertainties, such as norm-bounded, phase-bounded, and structured uncertainties, can lead to more robust and versatile stability conditions for complex systems.

What are the potential applications of the developed techniques in practical control system design and analysis

The developed techniques for analyzing robust stability under sectored-disk uncertainty have various practical applications in control system design and analysis. One key application is in the design of robust controllers for uncertain systems, where the proposed stability conditions can guide the selection of controller parameters to ensure stability in the presence of norm and phase constraints. These techniques can be particularly useful in industries such as aerospace, automotive, and robotics, where stability and performance are critical requirements. By leveraging the insights gained from the analysis of sectored-disk uncertainty, engineers can design controllers that exhibit robust performance and stability under a wide range of operating conditions.

Can the insights gained from the DW shell analysis be leveraged to study other matrix problems involving simultaneous constraints on norm and phase

The insights gained from the analysis of the Davis-Wielandt (DW) shell can be leveraged to study other matrix problems involving simultaneous constraints on norm and phase. By understanding the geometric properties of the DW shell and its relationship to matrix constraints, researchers can develop novel approaches for analyzing and solving matrix problems in various domains. For example, the DW shell analysis can be applied to study structured uncertainty in linear systems, investigate stability conditions for multi-input-multi-output (MIMO) systems, and design robust controllers for complex dynamical systems. By extending the DW shell analysis to different types of matrix problems, researchers can advance the field of robust control theory and enhance the stability analysis of diverse systems.
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